I'm trying to prove / disprove that the equation
$ 2 ^ x + 2 ^ y = 2 ^ z $
or $ x $, $ y $ and $ z $ are positive integers, only have trivial solutions. The obvious case where this is true would be for $ x = y $ but I'm not sure if solutions exist for $ x only $
I have the impression that there are no other solutions, but I do not know how one could officially prove it. Someone has advice on how to do this?
Sorry it's about a trivial issue / a known problem.